Answer:
Lowest resonant frequency: [tex]F_0=114Hz\\[/tex]
Wave Speed: [tex]v=198.36 \; m/s[/tex]
Explanation:
The lowest resonant frequency
For a fixed string a resonant mode is always a integer multiple of fundamental frequency (the lowest resonant frequency):
[tex]F_n =nF_0[/tex]
[tex]F_n[/tex] Is the frequency for a resonant mode
[tex]F_0[/tex] is fundamental frequency (the lowest resonant frequency).
Due to problem says that there's not intermediate resonant frequencies between 684 Hz and 798 Hz, i.e, they are consecutive resonant modes, equations are as follows:
[tex]684=nF_0 \; (1)\\798=(n+1)F_0 \; (2)[/tex]
Solving for n from equation (1) and replacing it in the equation (2)
[tex]790=(\frac{684}{F_0} +1)F_0[/tex]
And finally solving for [tex]F_0[/tex]
[tex]789=684+F_0 =>F_0=114 \;Hz[/tex]
Wave speed
For a fixed string the wave speed can be calculated by using
[tex]F_0=\frac{V}{2L}[/tex]
Where V is the wave speed and L is the string length.
So, using this equation and solving for V:
[tex]V=2*L*F_0 => V=2*(0.871 \; m)*114 \; Hz\\V=198.36 \;m/s[/tex]
It Is very important convert the string length unit to meters
